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Quadrilateral Sides Proportional Section

  1. Jun 26, 2006 #1
    Sorry guys, I thought it would be easy, but it is not.

    I have a random quadrilateral A,B,C,D, drawn and a random point M inside of it.
    Now, I want to draw a line passing through the point M and crossing the sides AB and DC at points P and Q.
    So far so good, but I want more, in addition the line is required to be constrained by this ratio of distances:

    AB:DC=AP:DQ
     
  2. jcsd
  3. Jun 29, 2006 #2
    I got some progress here.....

    Definition:
    If any line bisects the oppozite sides of a Qudrilateral A, B, C, D in points P and Q in such a manner the ratio AB:DC=AP:DQ holds than this line is called a Sweeping Line.

    Theorem by Milan Anthony Vlasak:

    "There is only one Parabola which is tangent to all Sweeping Lines of a Quadrilateral."

    Any objection to that ?
    You guys are mathematitions, I am a surveyor.

    Milan
     
  4. Jul 10, 2006 #3
    I can now EDIT the Definition:

    Edited Definition:
    If any line INTERSECTS the oppozite sides of a Qudrilateral A, B, C, D in points P and Q in such a manner the ratio AB:DC=AP:DQ holds than this line is called a Sweeping Line.
     
  5. Jul 11, 2006 #4

    0rthodontist

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    If you extend AD and BC until they meet at point R, then RM is a sweeping line.

    I don't think any parabola could be tangent to all sweeping lines of a quadrilateral.
     
    Last edited: Jul 11, 2006
  6. Jul 11, 2006 #5
    Orthodontist,

    If a line connects the midpoint of AB and the midpoint of CD than extention of this line will not generally pass the intersection point R of AD and BC.

    Such midpoint-midpoint line is still a sweeping line according to my definition of a sweeping line: AB:DC=AP:DQ

    (Nothing to do with collineation, cross ratio and Papus theorem)

    Milan
     
    Last edited: Jul 11, 2006
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